A geometric approach to Boolean matrix multiplication

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    For a Boolean matrix D, let r(D) be the minimum number of rectangles sufficient to cover exactly the rectilinear region formed by the 1-entries in D. Next, let m(D) be the minimum of the number of 0-entries and the number of 1-entries in D. Suppose that the rectilinear regions formed by the 1-entries in two n x n Boolean matrices A and B totally with q edges are given. We show that in time (O) over tilde (q + min{r(A)r(B), n(n + r(A)), n(n + r(B))})(1) one can construct a data structure which for any entry of the Boolean product of A and B reports whether or not it is equal to 1, and if so, reports also the so called witness of the entry, in time 0 (log q). As a corollary, we infer that if the matrices A and B are given as input, their product and the witnesses of the product can be computed in time (O) over tilde (n(n + min{r(A), r(B)})). This implies in particular that the product of A and B and its witnesses can be computed in time (O) over tilde (n(n + min{m(A),m(B)})). In contrast to the known sub-cubic algorithms for Boolean matrix multiplication based on arithmetic 0 - 1-matrix multiplication, our algorithms do not involve large hidden constants in their running time and are easy to implement.
    Originalspråkengelska
    Titel på gästpublikationAlgorithms and Computation : 13th International Symposium, ISAAC 2002, Vancouver, BC, Canada, November 21-23, 2002. Proceedings (Lecture Notes in Computer Science)
    FörlagSpringer
    Sidor501-510
    Volym2518
    StatusPublished - 2002
    EvenemangAlgorithms and Computation. 13th International Symposium, ISSAC 2002. - Vancouver, BC, Kanada
    Varaktighet: 2002 nov 212002 nov 23

    Publikationsserier

    Namn
    Volym2518
    ISSN (tryckt)0302-9743
    ISSN (elektroniskt)1611-3349

    Konferens

    KonferensAlgorithms and Computation. 13th International Symposium, ISSAC 2002.
    Land/TerritoriumKanada
    OrtVancouver, BC
    Period2002/11/212002/11/23

    Ämnesklassifikation (UKÄ)

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